Does anyone want an argument?

Here's a good one to get the ball rolling.

I do.

If you would just be so kind as to wait while I do some stretching exercises.

Deane

Right. I'm ready.

ejl, I think there should be some "discussion" before we decide on a topic.

ejl,

Godel is full of cobblers, seems to have an opinion on everything and is merely trying to subvert Neumann's Beweistheorie as established in 1927.

Well that can just be faulted on so many levels it's hard to know where to start.

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f(0,x2,…xn) = y(x2,…xn)
f(k+1,x2,…xn) = m(k,f(k,x2,…xn),x2,…xn).

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Well, I mean...

OK - here's a suggestion for a topic to "debate".......

"Does the modern Western approach to the retail merchandising of Christmas affect childrens view of the Christian Festival ?"

Godel's Incompleteness Theorem is the third greatest piece of human thought ever (the first two being my proposed system for seeding teams in the final stages of the world cup -- a piece of work that languishes unused on the desk of Sepp Blatter -- and Steven Toy's "Priniciples of Linguisitics Arising From Flaws In Chomsky's Transformational Grammar").

Matthew

I didn't come here for an argument.

Regards

Mike

Spending money I don't have on things I don't need.

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Originally posted by matthewr:
Godel's Incompleteness Theorem is the third greatest piece of human thought ever (the first two being my proposed system for seeding teams in the final stages of the world cup -- a piece of work that languishes unused on the desk of Sepp Blatter -- and Steven Toy's _"Priniciples of Linguisitics Arising From Flaws In Chomsky's Transformational Grammar"_).

Matthew

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Yep. Godel's incompleteness theorem should have finished philosophy for good. It sounds to me like most of the cutting edge thinking at the level of the purely abstract is being done in computer science departments.

Deane

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Originally posted by Deane F:
...It sounds to me like most of the cutting edge thinking at the level of the purely abstract is being done in computer science departments.

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Not hi-fi boards then?

Steve M

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Godel .... is merely trying to subvert Neumann's Beweistheorie as established in 1927.

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Actually Mike, this is wrong. Recent scholarship strongly suggests that Godel was responding to Carnap's (defective) Gabelbarkeitssatz of 1927 -- some references for you

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Yep. Godel's incompleteness theorem should have finished philosophy for good.

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Surely the opposite is true, Deane. Think about it: Suppose, contra Godel, that it did turn out to be possible to construct a syntactically complete axiomatic system of number theory. Wouldn't that have been about the strongest argument you could ask for that we are "just" mechanically computing results when we do math? In fact, the general consensus seems to be that the failure of the logicist program in the face of Godel's proof reinvigorated the philosophy of mathematics (Godel himself seemed to think to too, btw.)

Here's an interesting (if defective) attempt to use Godel's theorem to prove that we're not "just" machines.

Excuse me is this the five minute argument or the full half hour?

ejl,

Indeed, one could make the argument against the Gabelbarkeitssatz of 1927, but the issue I have with Godel is he never appears to be up to much. All he does is get back from work and spend all night posting his gauche viewpoints and theories on internet fora when noone really cares what he writes. We see it time and again. It's as if he just writes because he "likes the sound of his own voice", as it were.

I think he forgot to carry 1 on page 27!

Godel's incompleteness theorem proposes that any system of symbols capable of expressing arithmetic will be inconsistent and incomplete - not just mathematics.

A mechanistic theory of life and consciousness is too cold and stark to be contemplated.

I gree with Arnold Arnold that human beings are not machines. The biomechanical model of life is a crock.

Machines are designed for a purpose; I see nothing to suggest that human beings are the product of design. (But the fact that teleology runs through evolutionary theory like a backbone must be one of the greatest ironies of modern science.)

A machine can be dismantled into component parts and reassembled with no effect on the functioning of the machine. This is not possible with human beings.

If any component part of a machine malfunctions that component can (in theory) be replaced with an identical component with no effect on the functioning of the machine.

etc.

Deane

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Godel's incompleteness theorem proposes that any system of symbols capable of expressing arithmetic will be inconsistent and incomplete - not just mathematics.

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Not to argue with you Deane, but this doesn't make any sense man . Godel didn't prove that mathematics is inconsistent (that would have been a disaster); indeed, his proof presupposes that the axiom system at issue is consistent (otherwise his result doesn't follow, since you can formally "prove" anything from an inconsistency). Also, the proof doesn't concern systems of symbols (we can trivially express anything we want in symbols, including the Godel sentence "[R(q); q]"). Godel showed instead that there is no axiom system for arithmetic in which every (intuitively) true or false statement is provable.

I share your disinclination toward mechanism, but showing it's wrong isn't so simple. The points you mention are pretty easily answered (and, appearances to the contrary, modern evolutionary biology is NOT teleological -- all of the explanatory devices: gene mutation, reproduction, selection, etc., are straightforwardly causal).

Cheers,
Eric

Let us not mistake contradiction for argument.

cheers

Nigel

Deane,

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But the fact that teleology runs through evolutionary theory like a backbone must be one of the greatest ironies of modern science

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Well, it would ironic if that were true... but it ain't. The modern synthesis of evolutionary biology is anything but teleological, like The Eric sez.

Joe

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Originally posted by matthewr:
...and Steven Toy's _"Priniciples of Linguisitics Arising From Flaws In Chomsky's Transformational Grammar."

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There are some, however, who say it's not really much of an advance on Ian Robinson's quarter of a century old critique of Chomsky's Linguistics, The New Grammarians' Funeral

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A machine can be dismantled into component parts and reassembled with no effect on the functioning of the machine. This is not possible with human beings.

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Not true - what about a ring pull can

You putrescent bags of insect excrement, nodules of gaseous dogbreath; I ought to cosh you, you ... ...

... Oh sorry, I thought this was the abuse thread.

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Originally posted by bhazen:
... Oh sorry, I thought this was the abuse thread.

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It is if you are an American, goddamn you!!

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Originally posted by Stuart M:

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A machine can be dismantled into component parts and reassembled with no effect on the functioning of the machine. This is not possible with human beings.

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Not true - what about a ring pull can

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The collective peoples of the old Labour party into the New comes to mind !

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Originally posted by Deane F:

I gree with Arnold Arnold that human beings are not machines. The biomechanical model of life is a crock.

Machines are designed for a purpose; I see nothing to suggest that human beings are the product of design. (But the fact that teleology runs through evolutionary theory like a backbone must be one of the greatest ironies of modern science.)

A machine can be dismantled into component parts and reassembled with no effect on the functioning of the machine. This is not possible with human beings.

If any component part of a machine malfunctions that component can (in theory) be replaced with an identical component with no effect on the functioning of the machine.

Deane

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Not if it is dismantled and rebuilt by unskilled human beings.

Nime's Flaw will be found inscribed on my small pile of tomb rubble:

"If it aint broke, don't mend it!"

Nime

Duck!

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Originally posted by J. A. Toon:
Godel's theorem is a perfect example of precisely what is wrong with Mathematics in general, namely that it requires a stupidly complex and involved proof to demonstrate something that is so incredibly, blatantly, intuitively, immediately and obviously true to anyone who doesn't take Mathematics as gospel (if we abstract his theorem to its general conclusions).

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Although Godel's theorem does not seem counter-intuitive to me I must confess (speaking as a perpetual maths student whose intuition is, IMNSHO, usually second to none) that neither it nor any abstraction of it into other spheres of thought seems intuitively obvious to me. That is, unless you mean the distinction between mathematics and the meaning attributed to it, which is intuitively obvious to me but which is rather removed from Godel's theorem and cannot (as far as I can see) plausibly be described as a problem.

As for the idea that it is a failing of maths that complex proofs are needed to demonstrate the intuitively obvious, this misses the point as far as I am concerned. Since maths is, in a sense, all about tautologies one might use the metaphor of a journey that starts and ends at the same place: the journey itself is what matters, not the destination.

Since I do not understand what it means to "take mathematics as gospel" I cannot comment on that point.

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Mathematicians love to poke logical holes through non-mathematicians arguments, which is ironic seeing as mathematics is loaded with just as many if not more logical (and worse, philosophical) aberrations than virtually any subject, despite the fact that they might like to strenuously deny this reality.

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Do you know any such mathematicians? Actually, one needs a pretty advanced knowledge of logic, set theory and Heaven knows what else to understand any genuine problems with the foundations of maths. I certainly don't have an adequate level of knowledge, and am therefore not in a position to venture an opinion on the state of mathematics - which I suspect is true of the vast majority of maths graduates. To my mind, poking holes in an articulate forum bully's elementary logical howlers to reveal their stupidity or their disingenuousness - as I have had occasion to do at times - hardly bears comparison to this, so I see no is no irony at all.

As for philosophical problems, most of the ones I have come across have been meaningless (IMO) ones about the relationship between maths and its "meaning" or its "reality", which (IMO) should be confined to previous centuries - along with arguments about the number of angels that can dance on the head of a pin.

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And annoyingly, you'll never get anywhere in a debate with one because you'll spend forever trying to define your starting terms. Which in itself will not be neatly establishable.

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Hmmm... I think there is common ground in the fuzzy region between pedantic and meangingless. Whole episodes of Kilroy and Oprah have been predicated on the fact that people mean very different things by simple, widely used words such as, for example, "pornography". Participants in such debates can usually be trusted to have a raging, pointless argument rather than spend a little time establishing what their fellow participants actually mean.

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PS. Assuming I remember correctly:

Ø == {}

Okay... But, I also recall that:

Ø != {Ø}

Yeah right. Whatever. Nice big philosophical black hole there It might be "defined to be true" and it might "work mathematically", but this result doesn't actually mean anything sensible to me. How can something that solely "contains" that which is absolutely nothing be any different from nothing in the first place? Nothing is nothing. The "presence" (as such) of a non-presence is no different to a non-presence, despite however you may try to define it (especially as we are not considering something that used to be there, merely something that never was and never is there; i.e. isn't).

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For those who don't understand the notation, it says that the set that contains no elements is not the same as the set that contains the set that contains no elements.

To use an analogy, a box that comtains nothing cannot be said to be nothing simply because it contains nothing. And a box that contains a box that contains nothing does not contain nothing: it contains a box that contains nothing. So, clearly, a box that contains a box that contains nothing is not the same as a box that contains nothing.

I cannot see how that this, in itself, poses a problem of either meaning or philosophy. Problems of meaning in this context are conveniently meaningless to me anyway, so even if it did I would not consider it to be a problem. [or maybe I'm making this bit up so as to sound more argumentative].

As for the real question underlying this sort of thing [i.e. to do with Russell's Paradox - or nowadays problems in theories where Russell's Paradox is not a problem] I cannot begin to answer it. In the increasingly unlikely event that I get my poor, aching brain working at a level close to what was once normal for me - and I abandon my plan to avoid doing anything remotely mathematical when I finish my degree - I'll be delighted to do a PhD on the subject. You therefore have a very slight hope of receiving enlightenment from me in about seven years - but don't hold your breath...

I hope I have been satisfactorily argumentative. Sorry I didn't manage to criticise your logic or ask you to define your terms - you'll just have to be less logical and more vague in future.

[This message was edited by JeremyD on Thu 23 December 2004 at 3:25.]